This is a tough question to answer, but the simple answer comes down to the fact that rivers require rain to keep going. When it rains, excess water is soaked up by rocks. As the water runs out of the rocks and into the areas where rivers begin, the river gets more water.
However, many rivers have dried up before. Many have dried up in recent history. In the last couple of years, the demand for hydroelectric dams has also caused rivers to run dry, or at least parts of rivers to dry up. One of the best examples of this is the Mekong river.
This river runs through China, Burma, Laos, Thailand, Vietnam, and Cambodia. It starts in China, which has allowed China to channel the river into dams upstream that are causing the river to flood and dry a lot in the other countries.